# Chronological families: technical companion to C931

This companion states the mechanisms behind the synthesis. Equations describe supplied source constructions and explicitly marked diagnostics. They do not establish historical authorship or intention.

## 1. A labelled path and two measurements

For an ordered path with edge weights w_i and a held terminal T, a backward BC construction has x_i=T+sum_(j>=i) w_j. A regular chronology chooses begetting intervals; a cumulative chronology chooses the source-controlled lifespan/overlap measurement. The topology, weight rule and terminal are distinct inputs.

For two measurements u and v on a common path, let G_i=x_i(u)−x_i(v). Then:

G_i−G_(i+1)=u_i−v_i.

One terminal difference and the ordered local differences reconstruct the entire comparison field. This is the basis of the inherited regular/cumulative comparisons and the new22-node NT/MT field. In the latter, G_Jacob=1710. The native shared-name Arphaxad→Shelah edge measures140 in the NT display and35 in MT Rounded; the extra NT slot is Cainan. Explicit restoration produces NT70|70 versus MT35|130, retaining different measurements on the aligned graph. (C862–865,905.)

Rounding is a separate row operation. If Q(w_i)−w_i=e_i, the accumulated coordinate residual is the corresponding ordered suffix sum. Changing the rounded input field changes any comparison with a fixed field by the negative residual. This does not authorize rounding only the final total.

## 2. Lists are measured structures too

A category matrix retains information that its row and column margins lose. Both margins necessarily have the same grand total. In the Tishri matrix, a diagnostic2×2 circulation changes cells while preserving all margins; it is an information-loss witness, not an admitted ritual variant.

For the Esau purity/sex joint table [[440,50],[50,10]], equal490/60 margins require the off-diagonal balance50=50. The classifications remain distinct. The four margin measurements on the nine supplied category counts have rank3 because each partition sums to the same total. (C835–845.)

For weights (30,29,28,27,26,25,24,10), prefix accumulation and scale7 generate the complete forward festival path from1859BC. Reversing the declared sequence preserves the199 total but changes its prefixes. The two prefix sets intersect at0,59,140,199. These are consequences of the entire fixed sequence, not separate trials. A source-only packet and short regenerator are included. (C839–842,912.)

## 3. The NT grid, its count roles and common measure

Use civil x=1−BC or x=AD. The display rule is x(r)=−5−70r. It reproduces all78 rows of File43's schematic ledger. The primary77-row span is5390; the final AD65–135 row is an extension.

76 names Adam→Jesus have75 inter-name intervals:5250. Multiplying the name count by70 gives the separate5320 projection. Matthew's visible41 names give40 birth intervals,2800, or41 carrier slots through AD65,2870. These measurements do not overwrite each other. The God bookend is figurative. (C859–861.)

From a common origin,70k=49j iff k=7m and j=10m. The primary5390 span therefore has12 common boundaries at490-year spacing. The source1470|2450|1470 partition is3|5|3 such blocks. Reflection k→77−k closes the primary grid and swaps the BJ-marked indices21 and56. Grid closure is automatic; those named placements are supplied. The extended49-grid is a derived comparison, not a newly adopted BJ timeline. (C869–870.)

Ritual scaling7c lies on49Z,70Z or490Z exactly when7,10 or70 divides c. Applied to every prefix in both festival orders, the common490 positions are only count0 and140. Different anchors and event labels remain distinct. (C904.)

## 4. One-pass decimal evaluation

For an original nonnegative duration s=10^k times a decimal core whose final digit is nonzero, write the core digits from left to right as d_0,...,d_(l−1). The declared reversal is

I(s)=10^k sum_j d_j 10^j.

Thus trailing zeros remain placeholders. Given base10 and this policy, k and the core digits are computed from s; they are not additional primitive source data.

At anchor A, output A+I(D−A)=D+g(D−A), where g(s)=I(s)−s. For the paired File52c anchors differing by1400, put D−A=10n. The endpoint difference is g(10(n−140))−g(10n).

On the supplied nonzero-final-digit three-digit cases n=100a+10b+c:

| Branch in the supplied domain | Endpoint difference |
|---|---:|
| n≥240, b≥4 |990|
| n≥240, b<4 |1980|
|150≤n≤239 |1260+90a−90b−900c|
|144≤n≤149 |990−990c|

Fourteen supplied pairs use the long-core branches and sixteen use the short-core branches. The two remaining trailing-zero cases are Adam n410, difference7200, and Jacob n200, difference0. This partitions all32 source pairs and explains their64 existing one-pass endpoints. No second inversion or new statistical tally is involved. (C850–858.)

Moving source and anchor together leaves the duration unchanged. Moving the anchor while holding the source fixed changes the duration and possibly its decimal branch. This distinguishes whole-frame covariance from the paired-anchor comparison. (C903.)

## 5. Exact Keys and grid domains

Let s=hm on a declared grid hZ and let a Key be p/q in lowest terms. Its output remains in hZ iff q divides m. Therefore:

| Key | Ratio | Integer self-domain |5Z self-domain |
|---|---:|---:|---:|
|E|25/23|23Z|115Z|
|P|70/69|69Z|345Z|
|J|300/299|299Z|1495Z|

The intersections of their input self-domains are897Z and4485Z. Fractions outside a chosen display grid remain exact rational outputs.

For a fixed operation word with reduced prefix products p_j/q_j, every stage stays on hZ exactly when the input belongs to h*lcm(q_1,...,q_r)Z. The P/J witness is6877→6900→7000 under J then P, versus6877→20930/3→7000 under P then J. Same final product, different intermediate grid membership. These mixed words are diagnostics unless independently authorized. (C873–885.)

An anchored dilation is D_(k,a)(x)=kx+(1−k)a. Comparing the two compositions gives

D_(k,a)(D_(l,b)(x))−D_(l,b)(D_(k,a)(x))=(k−1)(l−1)(a−b).

This is a constant placement difference across the entire path. It is separate from the stagewise-grid issue. (C886–889.)

## 6. Endpoint pairs and retained biographies

For the six File63 pairs, the declared twelvefold rule is M=R+12(C−R)=12C−11R. Therefore ΔM=12ΔC−11ΔR. It generates the complete60,770,−3,−30 response field.

More generally, D=C−R and W_lambda=lambda*C+(1−lambda)*R retain both endpoints:

R=W_lambda−lambda*D; C=W_lambda+(1−lambda)*D.

The weighted Creation comparison uses lambda4/5; the macro heads use lambda12. Actual(14004,4114) and Rounded(14006,4106) share W=12026 but have spans9890 and9900, recovering different source pairs. (C892–894,901–902.)

For original outer span S and terminal biography l, uniform scaling gives partition k(S−l)|kl. Expanding the outer span while retaining the biography gives (kS−l)|l. The internal cuts differ by(k−1)l.

File63's P branch has S144900, k70/69 and retained l147. Its final147000 body contains999×147|147 at148936→2083→1936. Uniformly transforming the original147 would instead produce3430/23 and move the inner boundary to47958/23. The source's construction retains the biography; it does not uniformly transport that original cut. (C897–898,906.)

## Reading the result

These formulas reconstruct complete fields once the supplied inputs and roles are fixed. Primitive source choices include the weights, anchors, authorized operations, representation policies and retention decisions. Indices, digit registers, residuals and operator domains can then be computed. The finite loss witnesses support this working grammar without proving global uniqueness or historical causation. See the claim map for exact source sections and the evidence packet for all inputs, checks and reassessments.